The average least character nonresidue and further variations on a theme of Erdos

Abstract

For each nonprincipal Dirichlet character $\ddot{I}‡$, let $n_{\ddot{I}}‡$ be the least $n$ with $\ddot{I}‡(n)\notin \{0,1\}$. We show that as the average of $n_{\ddot{I}}‡$ over all nonprincipal characters $\ddot{I}‡$ modulo $q$ is $\ell (q)+o(1)$, where $\ell (q)$ denotes the least prime not dividing $q$. Moreover, if one averages over all nonprincipal characters of modulus at most $x$, the average approaches a particular limiting value 2.5350541804. We also prove a result of this type for cubic number fields: If one averages over all cubic fields $K$, ordered by the absolute value of their discriminant, then the mean value of the least rational prime that does not split completely in $K$ is another particular constant 2.1211027269.

19 pages